Use of log in likelihood
Below is an interesting question about likelihood from a thread in quora
Why do we always put log() in Maximum likelihood estimation before estimate the parameter?
The answer to this question is
\(log(x)\) is an increasing function. Therefore solving the following two problems gives the same result: \[ \begin{gather*} \underset{\theta}{max}\ f(x;\theta)\\ \underset{\theta}{max}\ log(f(x;\theta)) \end{gather*} \]
From above two equations, it seems there is no necessity to put log to solve the problem. The reason to put log is because most of the times it's faster to deal with sums than the products in the objective since it is more convenient to differentiate the sums than produces. For example, suppose we have \(n\) data points \(x_1,x_2,\cdots,x_n\) which are iid drawn from \(f(x;\theta)\) with unknown \(\theta\). MLE of \(\theta\) will solve the following problems:
\[ \begin{gather*} \underset{\theta}{max}\ \underset{i=1}{\overset{n}{\prod}}f(x_i;\theta)\\ \underset{\theta}{max}\ \underset{i=1}{\overset{n}{\sum}}log(f(x_i;\theta)) \end{gather*} \]
Two equations are equivalent, but it may take few additional steps to reach the same condition if we use the product